
Sparse Fast Fourier Transform :
The discrete Fourier transform (DFT) is one of the most important and widely used computational tasks. Its
applications are broad and include signal processing, communications, and audio/image/video compression.
Hence, fast algorithms for DFT are highly valuable. Currently, the fastest such algorithm is the Fast Fourier
Transform (FFT), which computes the DFT of an ndimensional signal in O(nlogn) time. The existence
of DFT algorithms faster than FFT is one of the central questions in the theory of algorithms.
A general algorithm for computing the exact DFT must take time at least proportional to its output size n.
In many applications, however, most of the Fourier coefficients of a signal are small or equal
to zero, i.e., the output of the DFT is sparse. This is the case for video signals, where a
typical 8x8 block in a video frame has on average 7 nonnegligible frequency coefficients (i.e., 89% of the
coefficients are negligible). For sparse signals, the Ω(n) lower bound for the complexity of DFT no longer applies. If a signal has
a small number k of nonzero Fourier coefficients the output of the Fourier
transform can be represented succinctly using only k coefficients. Hence, for such signals, we can find
DFT algorithms whose runtime is sublinear in the signal size, n.
We present here several new results for sparse Fourier transform:
 An O(k log n)time algorithm for the exactly ksparse case.
 An O(k log n log(n/k))time algorithm for the general case.
 An Ω(k log(n/k) /loglog n) lower bound for sample complexity.
Both algorithms improve over FFT, for any k = o(n).
Moreover, if one assume that FFT is optimal, the algorithm for the exactly ksparse case is optimal.
Under the same assumption, the result for the general case is at most one loglog n factor away from
the optimal runtime for the case of “large” sparsity k = n/log n.
We also have several follow up results
that improve the sample complexity of the algorithms. A summary of all the results can be found under the Theory section.
We further demonstrate that the sparse Fourier transfrom has impact on several practical applications in the areas of wireless communication, medical imaging, and computational photography.
The details of these applications can be found under the Applications section.
This research is supported in part by NSF and DARPA.



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